Fluorescence Recovery after Photobleaching (FRAP) is a versatile technique to study dynamic phenomena. Performing FRAP on a confocal laser scanning microscope documents the recovery process with high spatial resolution. This enables a consistent determination of the diffusion coefficient and the dimensionality of diffusion in calibration-free manner. Moreover, experiments representing multi-component diffusion can be analyzed as well, thus yielding the distribution of diffusion coefficients

Introduction

In the last 30 years, Fluorescence Recovery after Photobleaching (FRAP) has evolved as a versatile technique to determine diffusion coefficients of suitably labeled species in fields like pharmaceutical research, biophysics or polymer chemistry [1]. Basically, a FRAP experiment is realized by bleaching a certain area of a sample by short and intense laser irradiation. Afterwards, the diffusion of unbleached molecules from the surroundings leads to a temporal recovery of fluorescence intensity in the bleached region that is monitored with a highly attenuated beam. The diffusion coefficient can be deduced from the rate of recovery after suitable calibration. This procedure forms the basis for a variety of classical approaches to evaluate FRAP experiments [1].

Analysis of Spatially Resolved FRAP - Theoretical Background

When FRAP is performed on a confocal laser scanning microscope (CLSM), the recovery process can be followed with high spatial resolution on a µm-scale besides temporal resolution. Utilizing this aspect for a systematic analysis of FRAP data enables the estimation of the diffusion coefficient and the dimensionality of the diffusion process without any need for calibration [2].

Basically, each FRAP experiment makes use of the diffusion equation, which is also known as Fick's second law: see Formula (1)

where C represents the concentration of the substance under consideration, while D is the translational diffusion coefficient. Eq. (1) describes diffusion phenomena in an isotropic medium for the one-, two- or three-dimensional case. Solutions can be derived for certain initial and boundary conditions [3].

We focus on the following simple case which is relevant to FRAP experiments on a CLSM: see Formula (2)

Herein, r represents the generalized (radial) coordinate, and M denotes the total amount of the diffusing species in the three-dimensional case, while in the two- or one-dimensional case, it stands for the amount of substance per unit length or unit area, respectively. d symbolizes the diffusion dimension.
The shapes of the functions according to Eq. (2) are Gaussians with an e-^1/2-radius of √(2Dt). They broaden and become shallower with increasing time. Since the decrease of the prefactor with time depends on the dimensionality and the amount of diffusing substance, it is suited to estimate either of these parameters. On the other hand, the diffusion coefficient can be derived from the course of the widths.
If we consider FRAP processes, the situation is in essence just inversed, since bleaching takes away a certain amount of the fluorescent molecules, while the considerations above dealt with a local excess of a substance. This means no more than a change of sign combined with a baseline shift.

By utilizing a CLSM and an objective with a low NA value, it is readily possible to bleach geometries into the sample that correspond closely to the cases of diffusion from a plane (one-dimensional) or a line source (two-dimensional). Hence, the solutions of Fick's second law are applicable to FRAP processes too. One obtains Formula (3) where I represents the fluorescence intensity at the position r and the time t after bleaching. M now is formally a fluorescence intensity (per length or area for d = 2 or 1) corresponding to the amount of fluorophore destroyed by bleaching. w is the e-^1/2-radius of the Gaussian function and I₀ denotes the background intensity at r → ∞.

The quantity D appears in the prefactor and the exponent of the Gaussian, hence both terms can be used to determine it. Comparison of the exponential terms of Eq. (3) yields to Formula (4) nd therewith the possibility to deduce the diffusion coefficient by plotting w² vs. t for a series of intensity profiles obtained from images taken during the recovery process. A straight line with slope 2D passing through the origin should be expected.

On the other hand, we obtain an algebraic decay for the time dependence of A(t): see Formula (5)

As an alternative, Eq. (5) can be written in logarithmic form: see Formula (6)

This means that a plot of log A versus log t should give a straight line with a slope of -d/2, forming the basis for the experimental determination of the dimensionality of the diffusion process.

Analysis of Spatially Resolved FRAP - Practical Realization